$L $ denotes the second order partial differential operator having either the form
$$Lu=-\sum_{i,j} (a^{ij}u_{x_i})_{x_j}+ \sum _i b^iu_{x_i}+cu \tag{1}\label{1}$$ this is divergence form
The operator $L^*$, the formal adjoint of $L$ is
$$L^*v = -\sum_{i,j} (a^{ij}v_{x_j})_{x_i}- \sum _i b^iv_{x_i}+(c-\sum_i b^i_{x_i})u \tag{2}\label{2}$$
My question is how $\eqref{2}$ derives from $\eqref{1}$.
The adjoint operator definition is $<Tx,y>=<x,T^*y>$ and I am confusing here we use $L^2$ norm or $H_0^1$ norm and what is the dual of $H_0^1$.
It is formal in the sense that $$ \int_\Omega (Lu)(x) v(x) dx = \int_\Omega (L^*v)(x)u(x)dx $$ for all $u,v\in C_c^\infty(\Omega)$. The formula for $L^*$ can be obtained by integration by parts. Under assumptions on the coefficients and the domain, one can make this rigouros (i.e., $L^*$ is adjoint operator of $L$).